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%I A158390 #2 Oct 12 2012 14:54:56
%S A158390 1,1,-1,-4,-1,1,-36,9,5,-1,576,64,-61,-5,1,14400,-1600,-1045,105,14,
%T A158390 -1,-518400,-32400,29088,1593,-385,-14,1,-25401600,1587600,1079200,
%U A158390 -63481,-11521,579,30,-1,1625702400,65028096,-55637056,-2062912,523001,16045
%N A158390 A triangle of matrix polynomials: M(d)=Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]]*Transpose[Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]]].
%C A158390 Row sums are:
%C A158390 {1, 0, -4, -23, 575, 11873, -520517, -22809194, 1633567982, 122963894441,
%C A158390 -13226887049109,...}.
%C A158390 Example matrix:
%C A158390 M(3)={{1, 2, 3},
%C A158390 {2, 4, 0},
%C A158390 {3, 0, 0}}
%F A158390 M(d)=Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]]*Transpose[Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]]];
%F A158390 out_(n,m)=coefficients(characteristicpolynomial(M(n),x),x)
%e A158390 {1},
%e A158390 {1, -1},
%e A158390 {-4, -1, 1},
%e A158390 {-36, 9, 5, -1},
%e A158390 {576, 64, -61, -5, 1},
%e A158390 {14400, -1600, -1045, 105, 14, -1},
%e A158390 {-518400, -32400, 29088, 1593, -385, -14, 1},
%e A158390 {-25401600, 1587600, 1079200, -63481, -11521, 579, 30, -1},
%e A158390 {1625702400, 65028096, -55637056, -2062912, 523001, 16045, -1563, -30, 1},
%e A158390 {131681894400, -5267275776, -3616612416, 138811968, 28130697, -981153, -75500, 2167, 55, -1},
%e A158390 {-13168189440000, -365783040000, 301206312000, 7960416400, -2038537300, -48112905, 5258465, 99136, -4851, -55, 1}
%t A158390 Clear[M, T, d, a, x, a0];
%t A158390 M[d_] :=Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]]*Transpose[Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]]];
%t A158390 a0 = Table[M[d], {d, 1, 10}];
%t A158390 Table[Det[M[d]], {d, 1, 10}];
%t A158390 Table[CharacteristicPolynomial[M[d], x], {d, 1, 10}];
%t A158390 Flatten[a]
%t A158390 Join[{1}, Table[Apply[Plus, CoefficientList[ Expand[CharacteristicPolynomial[M[n], x]], x]], {n, 1, 10}];
%K A158390 sign,tabl
%O A158390 0,4
%A A158390 _Roger L. Bagula_ and _Gary W. Adamson_, Mar 17 2009

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